The 0/1 Knapsack Problem is a classical combinatorial optimization pro

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Module 2: Fundamental Algorithmic Strategies

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The 0/1 Knapsack Problem is a classical combinatorial optimization problem. Explain how this problem can be solved using:
(i) Brute-force method
(ii) Greedy method
(iii) Dynamic Programming
(iv) Branch-and-Bound
For each approach, explain the algorithm, time complexity, and limitations. Highlight why the greedy method may not always yield the optimal solution.

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PreviousAnswer the following: (i) Compare dynamic programming and greedy algorithms in terms of their problem-solving approaches, solution spaces, and efficiency. (ii) Discuss scenarios where dynamic programming is necessary because greedy strategies fail. Use suitable problem examples to support your explanation. (iii) Explain how overlapping subproblems and optimal substructure are utilized in dynamic programming.NextConsider a TSP with 4 cities A, B, C, D and the following distance matrix: | | A | B | C | D | |---|---|---|---|---| | A | 0 | 10 | 15 | 20 | | B | 10 | 0 | 35 | 25 | | C | 15 | 35 | 0 | 30 | | D | 20 | 25 | 30 | 0 | Answer the following: (i) Use the branch-and-bound algorithm to find the shortest possible route that visits all cities exactly once and returns to the starting city. (ii) Show the branching steps and calculate lower bounds for each partial solution. (iii) Identify how the bounding helps prune the search tree.